I have a (supposedly) calculus problem that I just can't seem to figure out. Basically, I'm trying to understand why alternative kinetic energy formulation does not yield the same equations of motion in problem 11 of Goldstein's Classic Mechanics 3 edition.

The text of problem is following:

Consider a uniform disc that rolls without slipping on a horizontal plane. A horizontal force is applied to the center of the disc and in a direction parallel to the plane of the disk.

However, if I formulate kinetic energy as [tex]T = m * (v_x/cos(theta))^2 / 2[/tex], everything in the equation seems to change with the additional dependence on theta (the angle of disc orientation on the xy plane).

Is there anything wrong with using this alternative kinetic energy formulation (except that it blows up on [tex]theta = pi/2[/tex])?

Oh, I think I figured it out. The virtual displacements [tex]∂q_i[/tex] are not independent in these coordinates, therefore if my [tex]T[/tex] depends on both [tex]x[/tex] and [tex]θ[/tex], I have to take both in account to create an equation of motion.